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Functionalizing graphene was recently shown to have a dramatic effect on theelectronic properties of this material. Here we investigate spatial ordering ofadatoms driven by the RKKY-type interactions. In the ordered state, whicharises via a Peierls-instability-type mechanism, the adatoms reside mainly onone of the two graphene sublattices. Bragg scattering of electron waves inducedby sublattice symmetry breaking results in a band gap opening, whereby Diracfermions acquire a finite mass. The band gap is found to be immune to theadatoms' positional disorder, with only an exponentially small number oflocalized states residing in the gap. The gapped state is stabilized in a widerange of electron doping. Our findings show that controlled adsorption ofadatoms or molecules provides a route to engineering a tunable band gap ingraphene.

Peierls-type Instability and Tunable Band Gap in Functionalized Graphene